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We consider the problem of parameter estimation in a partially observed linear Gaussian system with small noises in the state and observation equations. We describe asymptotic properties of the MLE and Bayes estimators in the setting with…
We consider a high-dimensional mean estimation problem over a binary hidden Markov model, which illuminates the interplay between memory in data, sample size, dimension, and signal strength in statistical inference. In this model, an…
Discovering compact governing equations from experimental observations is one of the defining objectives of quantitative science, yet practical discovery pipelines routinely fail when measurements are noisy, relevant state variables are…
The sparse modeling is an evident manifestation capturing the parsimony principle just described, and sparse models are widespread in statistics, physics, information sciences, neuroscience, computational mathematics, and so on. In…
Estimation of Markov Random Field and covariance models from high-dimensional data represents a canonical problem that has received a lot of attention in the literature. A key assumption, widely employed, is that of {\em sparsity} of the…
Our ability to predict, control, and ultimately understand complex systems rests on discovering the equations that govern their dynamics. Identifying these equations directly from noisy, limited observations has therefore become a central…
Enforcing sparse structure within learning has led to significant advances in the field of data-driven discovery of dynamical systems. However, such methods require access not only to time-series of the state of the dynamical system, but…
Sparse system identification is the data-driven process of obtaining parsimonious differential equations that describe the evolution of a dynamical system, balancing model complexity and accuracy. There has been rapid innovation in system…
We consider a Kuramoto-Shivashinsky like equation close to the threshold of instability with additive white noise and spatially periodic boundary conditions which simultaneously exhibit Turing bifurcations with a spatial 1:3 resonance of…
Bayesian methods have been very successful in quantifying uncertainty in physics-based problems in parameter estimation and prediction. In these cases, physical measurements y are modeled as the best fit of a physics-based model…
First principles modeling of physical systems has led to significant technological advances across all branches of science. For nonlinear systems, however, small modeling errors can lead to significant deviations from the true, measured…
A state-space model is a statistical framework for inferring latent states from observed time-series data. However, inference with nonlinear and high-dimensional state-space models remains challenging. To this end, an approach based on…
The discovery of the theory of compressed sensing brought the realisation that many inverse problems can be solved even when measurements are "incomplete". This is particularly interesting in magnetic resonance imaging (MRI), where long…
The Kuramoto model has been introduced to describe synchronization phenomena observed in groups of cells, individuals, circuits, etc. The model consists of $N$ interacting oscillators on the one dimensional sphere $\mathbf{S}^{1}$, driven…
Fault diagnosis plays an essential role in reducing the maintenance costs of rotating machinery manufacturing systems. In many real applications of fault detection and diagnosis, data tend to be imbalanced, meaning that the number of…
Data assimilation is a central problem in many geophysical applications, such as weather forecasting. It aims to estimate the state of a potentially large system, such as the atmosphere, from sparse observations, supplemented by prior…
This chapter provides various perspective on an important challenge in data assimilation: model error. While the overall goal is to understand the implication of model error of any type in data assimilation, we emphasize on the effect of…
This work is concerned with uncertainty quantification in reduced-order dynamical system identification. Reduced-order models for system dynamics are ubiquitous in design and control applications and recent efforts focus on their…
The interpretability of machine learning, particularly for deep neural networks, is crucial for decision making in real-world applications. One approach is replacing the un-interpretable machine learning model with a surrogate model, which…
Data assimilation (DA) provides a general framework for estimation in dynamical systems based on the concepts of Bayesian inference. This constitutes a common basis for the different linear and nonlinear filtering and smoothing techniques…